I had meant to mention in the preceding post that Peikoff 1964 notes that not all classical philosophers subscribed to a metaphysical distinction between the necessary and the contingent. He helpfully mentions John Scotus Erigena, Spinoza, and Hegel.
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Links to the sections of this essay so far:
Plato
Aristotle I II
Kant I II III
Conventionalism I II III

I concur with the distinction Merlin draws between physical and formal necessity in the preceding post. That’s a good example from mathematics, and I should note additionally that (i) it is a fact—ascertained in the way one does for mathematics—that there are some continuous functions that are nowhere differentiable, and it remains a fact even if it is the case that there simply is nothing physical to which some such function applies and that (ii) we find great success in technology and in extending comprehension of the physical by applying many functions, each one both continuous and differentiable, to electricity, to fluids, and to solids, yet understanding perfectly well that such things are discontinuous at small enough scales.
SL, I should not want to equate the physical with the metaphysical. When Rand claims that only living things can have values or when philosophers from time immemorial say nothing comes from nothing, those claims are consonant with modern physical science, but the claims are made in what I’d call a metaphysical perspective, not a scientific one.
In his 1967 essay “The Analytic-Synthetic Dichotomy” Peikoff has a section on the traditional distinction within metaphysics between necessary and contingent facts (and how this feeds into the A-S distinction). The meaning of metaphysical necessary/contingent has changed over the centuries, but there is family-descendant resemblance under the continuing distinction. Peikoff did not think such a distinction is correct to make within metaphysics. However, he there drew a distinction between the metaphysical and the manmade (in tune with Rand’s later elaboration). Human free will is the root fact for this distinction. Unfortunately, Peikoff and Rand thought that the rule of Identity in metaphysics entailed complete determinism throughout metaphysics as contrasted with the realm of free will. Furthermore, Rand thought that such metaphysics rightly constrains (a bit) what physical science might find, but that the reverse flow does not soundly occur. That is, she thought metaphysical fundamentals could not be changed in light of advances in science. So for example, the development of chaos theory in the classical regime of physics (starting in the 1970’s as I recall) and the distinction within physics between a classical system in its regular regime as opposed to being in its chaotic regime could not suggest any reformation of general metaphysics. Really, the total determinism that Rand-Peikoff attached to metaphysics under identity was an inheritance from modern physics (Laplace et al.) and is not properly part of right metaphysics, rather should be left open for physics to settle. In his book OPAR, Peikoff does acknowledge that when it comes to value theory, biology supplies the characterized phenomena, pertinent for philosophical fundamentals concerning value.
In his dissertation, Merlin, Peikoff included Blanshard’s books The Nature of Thoughtand Reason and Analysis. He does not cite the former in his text or notes. He cites and makes specific explicit use of the latter from its pages 252–54 and 271–75. The former stretch lays out the traditional view that necessity (the one, as it happens, to be most often sainted by philosophers traditionally) arises only at the level of universals and essences; discerned at the level of conception, not perception. The latter stretch concerns conventionalist theories of logic.
Merlin, I’ve inclined to the view of logic put forth by Rand (1957) and Branden (c. 1968) and Peikoff (1967, 1991) in their orientation towards logic as tool for successful thinking. (I reject Rand’s definition of logic in its differentia. I expect she was misled by a remark in Aristotle’s Metaphysics, which seems oblivious to his great achievement, theory of the syllogism, in Prior Analytics.) It has seemed plain that on the Objectivist orientation towards logic, material implication should not be incorporated. A lot of other thinkers have thought material implication off the mark for deficiency in the relevance factor, as had Blanshard. They developed Relevance Logic (also called Relevant Logic) as replacement for classical modern logic, and I think that the way to go and a way consonant with Objectivism also. I have books telling the history, concerns, and purposes that brought on material implication, but I’ll have to open them. I’ll let you know on your blog what I find.

PNC Ground Shifts to the Side of the Subject– Conventionalism III
To set myself the task “weed this patch of periwinkles” I may need to use language. The two popular weeds there at this season are dog violets and a native vine I don’t know the name of. Getting to the nub of that weed-vine among the thicket of periwinkle vines and pulling out the former without pulling out the latter is a challenge. Names and language do not seem to be enlisted in executing the task; they enable only my report of this work. The weed-vine and the periwinkle are of different leaf shape and color. Tug gently on the end of the weed-vine reaching for the sun. You won’t be able to see the weed-vine you’re tugging but a few inches before it disappears (leafless in this portion of it) among the thicket of periwinkle vines hugging the earth and putting down their roots continually along their way. But as you tug on the weed-vine, you’ll be able to find with your other hand that single vine being tugged. It is tightly tensed and in synchrony with any rhythm of tugs you apply with the other hand. Repeat from there, and eventually you arrive at the nub of the weed-vine and pull out that vine by the root.
Pause at a step in which you have the single obscured weed-vine in each hand. Pull with the one hand, feel the pull in the other. That is a perceived connection between two distinct events. At this point, philosophers from Plato and Aristotle to Hume and Kant stick up their noses. Not Locke.
That applied force can be conveyed along a vine is a physical necessity. That different things in general (as example, weed-vines and periwinkle vines) are not same things is another type of necessity, logical necessity, however neatly it coincides with physical necessity. Logical necessity holds unconditionally and in all contexts. What I’ve called physical necessity is traditionally taken to be necessity under some sort of limiting conditions, and this necessity has been called a contingent connection, reserving necessary connectionfor logical (and other formal) necessity. The real distinction, I think contrariwise, should be in what aspects of things we are accessing and the different ways these two aspects are accessed.
Peikoff 1964 points out that Locke avoided the contingent/necessaryterminology. Locke instead applied probable/certainto the division. We have seen in my section Aristotle II that Locke maintained we have by sensory perception instances of the general fact that different things are not same things and that a thing is never both A and not A at the same time and in the same respect. Philosophers, including Peikoff in 1964, are correct to fault Locke’s blurring under probable/certaina clear understanding that ampliative inductive generalizations over perceived instances do not suffice to land the absolute necessity in general principles of logic or pure mathematics. Peikoff notes on page 218 the parallel criticism in Hume’s famous dictum that we do not find in sense perception any necessary connection between distinct events (distinct impressions,in Hume’s own parlance and perspective). Countering Hume’s quandary, Kant attempted a radical subject-sided formulation of necessities such as the necessity in a principle of causality, a reformulation in which Kant would have objective temporal order of distinct events get the necessity of that order from a necessity of causal structure demanded by human mind. (Cf. Peikoff 2012, 32–33.)
Locke had fogged up by his softening of the distinction between (i) the physical necessities one can sense and manipulate with the weed-vine in one’s hands and (ii) formal and metaphysical necessities. Nevertheless, I maintain Locke right in taking (i) to be the driver of (ii) and not the other way around, as philosophers from Plato to Kant and beyond would have it. British empiricism has its good sense even if it was never good enough.
Locke was not really of one mind in this. Peikoff lays out an opposite strand also inAn Essay Concerning Human Understanding: IV 3.31, 4.6, 4.8, 9.1, 11.13–14. “What is Locke doing in such passages as these? He is now contrastingeternal truths and existential truths. The former are to be discovered only by ‘the examining of our own ideas’, and ‘concern not existence’ . . .” (222). Peikoff points out that the likes of platonist Cudworth or Leibniz had also maintained such a division, but for them consideration of our own ideas accesses the eternal truths as immutable relations in the divine understanding. Eternal truths such as the laws of identity and noncontradiction, as well as the essences of existing things, are givens to the human mind, independently of our self-examinations accessing them. But for an empiricist such as Locke, rejecting that rationalism, and joining considerable nominalism (the conceptualist wing of nominalism) concerning universal ideas to the empiricism, the divide between matters of fact and the eternal, formal truths can make conventionalism concerning the ground of logic “almost inevitable” (223).
The leading German spokesman for conventionalism in science, geometry, and logic in the early years of the twentieth century was Hugo Dingler: “The application of the law of contradiction rests on my free will. . . and this is just what is called a stipulation [Festsetzung]” (1919, 14-15; quoted in Carus 2007, 120n14). “There is no other way to guarantee the general validity of a law other than its stipulation by the will” (1919, 13; Carus 119). Peikoff would not likely have known much about this history in 1964, much beyond, that is, what Popper wrote against it in his 1934 The Logic of Scientific Discovery. I want to point it out because although Dingler rejected as unfounded Kant’s basis of the necessity in geometry as arising from synthetic a priori judgments and Kant’s picture of how certain laws are a priori conditions of the possibility of any experience (Wolters 1988). Dingler is nonetheless a redo of Kant, of the first Critique,with conscious choice (of alleged conventions) replacing Kant’s mandatory structure in any sensory intuition and in any conceptualization of things external to mind. Though crucial, fundamental organization of mind on Dingler’s view is voluntary, and although Kant would shake his head over such free play as that, it remains that the organization is an a priori condition for the possibility of any experience or knowledge.
Carnap will resist such radical conventionalism in the 20’s and 30’s. I’ll return in the next installment to the course of Logical Empiricism and the role of (still overextended) conventionalism in their characterization of logic and in the characterization by Dewey and by C. I. Lewis. I expect to yet dig into the fate of conventionalism concerning logic to the present day. Jumping out of chronological order, just now I want to be sure to mention—to show that conventionalism in logic remains a current and a concern in philosophy today—the section 6.5 “Logical Conventionalism” in Theodore Sider’s Writing the Book of the World (2011 Oxford).
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Carus, A. W. 2007. Carnap and Twentieth-Century Thought – Explication as Enlightenment.Cambridge.
Dingler, H. 1919. The Foundations of Physics: Synthetic Principles of Mathematical Natural Philosophy.Union for Scientific Publishing, Berlin and Leipzig. (In German.)
Peikoff, L. 2012. The DIM Hypothesis.New American Library.
Wolters, G. 1988. Hugo Dingler. Science in Context2(2):359–67.

GB, two architects who have been inspired expressly by Ayn Rand are:
John Gillis https://www.architecturaldigest.com/story/stamford-connecticut-john-gillis
Peter Cresswell http://organonarchitecture.co.nz/CLIENTS/Design_Consultation_Brochure.pdf

Veritas, why can't Dr. Binswanger and others working within Rand's system take consciousness as awareness to be a first-person standpoint and fundamentally in contrast to a third-person standpoint such as when they say that consciousness is some sort of brain processing, which is to say physiological, which is to say physical? In other words, couldn't one say that in certain sorts of contexts it is sensible to say consciousness is physical and in other sorts of contexts consciousness is in stark contrast to the physical?
Does what Binswanger writes rule out that option for him?

This question of William’s has been very fertile.
In his Intermediate Logic (1997), David Bostock argues there is a circumstance to be mentally entertained, a circumstance that has as its result that Modus Ponens would not be a valid rule of inference. Because of that result, he concludes that that circumstance is illogical (350–54). Because the circumstance pertains to empty referent for referring terms in logical relations, I incline to think all the more that knowing Modus Ponens is contained within and should be isolated within knowing consciousness is identification, where logic is understood to be a certain subdivision of verbal consciousness as identification. What then are the particulars of how we know the logic subdivision of such consciousness? Among those particulars should be how we know validity of Modus Ponens.
As observational givens that William mentions, I think for Rand’s epistemology as it leads to knowledge of logical principles, we must start with verbal reports of observation such as “this pen still has ink” and “this board is less bowed than that one.” How we know such observational reports of ours are true when they are true is one layer of epistemology. How we know logic is a further layer of epistemology, and how we know Modus Ponens has to be part of that further layer of hows.
Rand gestured in her epistemology that there are significant relations between (i) observation and elementary conceptual processes concerning observations and (ii) processes of induction and deduction (ITOE 28). That variety of induction would be most plausibly the sort of induction we know as abstractive induction (also known as intuitive induction). There was an attempt to expand on this gesture of Rand’s in the first chapter of David Harriman’s book The Logical Leap (2010), but it discussed the relation of observation, conceptualization, and ampliative induction, not abstractive induction. And the latter is what is relevant to how we know deductive logic.

The distinguished linguist James McCawley wrote:
“I know of no one in linguistics who accepts the idea that the structure of one’s native language imposes limits on what thoughts one can think.”
For example, “all languages have simple ways of referring to the future, but they don’t necessarily use tenses of verbs for that purpose. Speakers of English are no better at thinking in terms of the future than are speakers of Chinese, which has no tense forms at all, nor any worse than speakers of Kikuyu, which has distinct near future and remote future tenses.”
These are excerpts from a letter on page 6 here.
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Joseph Conrad and Ayn Rand were two excellent novelists in the English language, even though it was not their native language. Rand was also able to express philosophical ideas well in English. However, among people I've personally known, I've found that if English was not their native language, they have trouble understanding and expressing philosophical ideas with precision in English. So although grasp and expression of some ideas might be inherently more direct or more roundabout in one language or another as one navigates reality with it, I suggest that one's greatest competence for grasping reality is, for most of us, our own native language.

PNC Ground Shifts to the Side of the Subject – Conventionalism II
Logical empiricists rejected Kant’s synthetic a priori as a class of propositions, and they rejected as well Kant’s role of intuition in arithmetic and geometry. All a priori propositions were analytic for these twentieth century philosophers. Having taken that position, they took some modern philosophers before Kant, such as Leibniz or Hume, to have been on the right track; they saw Kant as a derailment. I should note that none of these twentieth century philosophers giving a significant nod to convention in logical principles, such as PNC, were epistemological skeptics. They found attractive Hume’s wall between abstract reasoning and matters of fact, which was similar to epistemological dipoles of their own. They could applaud his clipping wings of metaphysics. They treasured mathematics and modern empirical science, and they did not give an inch to religion.
L. E. J. Brouwer threw a Kantian intuitionist spanner into the logical empiricist program in the late 1920’s. He formulated an intuitionist, constructivist, and finitary conception of mathematics which implied the invalidity of a significant portion of classical mathematics that had been developed by that time. Carnap proclaimed legitimate in broad perspective both the Brouwer system and classical mathematics by characterizing them as devoid of factual content and by leaving them to win the day according to which could best serve the formal deductive needs of empirical science.[1]
The selection between Brouwer’s intuitionist mathematics and classical mathematics for the boosting of empirical science is not arbitrary. What is better suited or best suited to some end, such as boosting empirical science, is a matter of objective fact. The following, I notice, share nothing with the arbitrariness that enters convention (just as the need for a left-or-right-side driving rule shares nothing with the arbitrariness that enters the choice of which side): what is more rather than less convenient, what is more rather than less simple, and what is more rather than less practical. One slip into a subject-siding error in this neighborhood would be to say that because a model or a theory is more convenient or simple or practical, it is more likely to be true (or, stepping with Protagoras, it is what truth is).
Within pure, unapplied geometry, it is false to say there is no correctness or incorrectness concerning hyperbolic geometry, elliptical geometry, and Euclidean geometry; all three are true within the discipline of pure mathematics. The impulse to consider these geometries as somehow not only distinct from, but as opposed to each other within pure geometry is wrong thinking. There is some truth to the conclusion of Poincaré and, later, the logical empiricists that question of which, between hyperbolic, elliptical, or Euclidean, within pure geometry is true (concerning their differences, not their commonalities) is a meaningless question. The context, I say, that makes such a question meaningless is a context in which there are facts, albeit facts not empirical. So Carnap was wrong to say additionally that purely formal disciplines and their systems are themselves devoid of factual content. It is misleading to confine usage of fact to empirical fact, just as it would be misleading to confine circumstance or form to empirical circumstance or empirical form.
(My own view is that there are formal lays of the physical and ways of ours with the physical that are not empirical lays of the physical and not our empirical ways with the physical. Specifics are reserved for my book in progress.)
Henri Poincaré died in 1912. He lived long enough to assimilate modern geometry and special relativity, and the Minkowskian geometric character of SR spacetime into his epistemological views on physics and on mathematics. Poincaré did not live to see the advent of general relativity (1915), with its condensations of the principle of inertia into spacetime geometry and gravitational force into inertial force, its spacetime structure affecting motions of mass-energy, and its distribution of mass-energy dictating spacetime structure, all at play in one super-fertile physical equation, Einstein’s field equation. Poincaré had over-extended the role of convention in both physical theory (kinematics and dynamics, including SR) and modern geometry. Those over-extensions have been soundly refuted, even without setting that much physics and physical geometry within general relativity. Those repudiations aside, general relativity was utterly devastating to the roles Poincaré had purported for conventions in physics and physical geometry.[2] The logical empiricists sometimes situated conventions in logical truths in ways self-consciously similar to ways Poincaré had (mistakenly) thought he found sturdy niches for convention in physics and geometry.
That does not mean that every such mimic of Poincaré mistaken. I should say only beware, for separating what is conventional and what is not is not always easy, within one’s present context of knowledge. But the more important point I want to make for our present examination of possible connection of PNC ontological and epistemological character from Kant to conventionalism is that Poincaré held a generalized version of Kant’s synthetic a priori status for arithmetic, geometry, and fundamental mechanics. This generalized version with its niches for convention possessed those niches only due to advances in mathematics and science since the time of Kant. No shifting of ontology to the side of the subject nor deflation of ontology by Kant, in his specific ways, seems to be required for Poincaré to have made his conventionalist moves. And the logical empiricists made their conventionalist moves on logical truths, including PNC, without any reliance on, indeed in flat denial of the Kantian class of the a priori that is also synthetic. Kant’s critical philosophy further sealed the tomb of logical ontologism, but in my assessment thus far, Kant prepared no ground and planted no seeds for the spring of twentieth-century conventionalisms in the character of logic or its applications.
But what about Kant via tributaries from neo-Kantians (viz., Marburg ones) into logical empiricism?
(To be continued.)
[1] Friedman 2010, 669–76.
[2] Ben-Menahem 2006, 40–68; Friedman 2010, 642–64; Gray 2013, 525–33.
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Ben-Menahem, Y. 2006. Conventionalism. Cambridge.
Friedman, M. 2010. Synthetic History Reconsidered. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.
Gray, J. 2013. Henri Poincaré – A Scientific Biography. Princeton.

SL,
I don’t think that Rand’s character of categories requires that all concrete existents belong to one of those four categories and not to the other three. Those four are entity, action, attribute, and relationship. Firstly, one could take angular momentum, for example, to be truly an action but also an attribute, and a relation. Her way with categories is simply different than Aristotle’s way of absolutely unique categorization of a thing. (Please, anyone, correct me if I’m wrong about that point on Aristotle.) But secondly, and pertinent to the spacetime/mass-energy characterizations into Rand’s categories, it has seemed to me that anytime a concrete existent is understood as a system, one can rightly take it as an entity. For example, in AS 1016, Rand takes the solar system to be an entity. One could also take it as “this particular matter with such-and-such orbital angular momentum about the sun, also this other particular matter with such-and-such other orbital angular momentum about the sun, also . . .” Then we’ve a summation of actions of entities, not an entity. I’m comfortable with that sort of multiple categorization of a thing where it is true to a thing.
So I’d think it fine to take spacetime in its global structures to be an entity even if locally it were not an entity, but a relationship. And in GR we’d take this room I’m in to be, over very short, shorter, . . . periods of time, as asymptotically an inertial frame of motion. I’m fine with taking the space contained by these walls and containing me as not only an entity containing other sorts of entities, and the spacetime entity I’ve here as asymptotically of zero curvature; but as well, as a collection of a certain kind of relationships between other sorts of entities.
Probably the most important multiplicity of category that Rand employed was the ontological status of mind. As an operating system, an instrumentation and control system, it’s an entity. But it’s also a process and activity, that is, it falls in the category action.
The utility of Rand’s categories seem somewhat like the utility of a certain easy network understanding of a thing, a hand-over-hand sort of comprehension of a thing (although this easy network is an alternation staying outside entity): “A pear is a kind of fruit which is a part of a pear tree which is a kind of plant which (with others) is a part of the biosphere.” (30)
None of this entails, I should mention, that entity is a category not having primacy over other categories of existents, primacy in acquisition of language, in conceptual dependencies, and in ontological relations.
[I notice that Kant's categories do seem to require no dual memberships. Perhaps that is because they are lifted from distinct logical forms of judgment. The latter could reflect basic ontological standings (contrary to Kant's conception of their ultimate source and justification). Kant's categories seem, however, less readily useful than a freer and more accessible set of categories such as Rand's.]

First Earth-Based Radio-Wave Image of Galactic Black Hole (4/10/19)
In general relativity, including in its combine with quantum field theory at the event horizon of a black hole (Hawking radiation), mass-energy is one thing and spacetime with all its curvatures is another thing. Mass-energy is an entity. Distribution of mass-energy in spacetime determines how spacetime will curve. A thing susceptible to such a dynamics is an entity, I'd say, or at least it is some sort of concrete existent. So I think of spacetime---even empty spacetime, i.e., even spacetime if it had no vacuum energy---as an entity. Spacetime curvature is a causal factor in how mass-energy moves. This too supports the classification of physical spacetime as an entity.
In talking of entity and of concrete existent, I'm talking of some philosophical, metaphysical categories, specifically some categories in Ayn Rand's metaphysical scheme. That sort of broad framework is useful for assimilating and keeping somewhat unified all the areas of one's experience and learning. Methods of successful science are in part from rational philosophy (rational epistemology) down the ages as the discipline of philosophy assimilated and analyzed such success in science and mathematics as had been attained. However, in the mature sciences such as modern relativity physics, astrophysics, and astronomy, additional methods for success have also been forged by some scientists themselves (under their epistemology thinking cap, we might say) as they hunted what is in nature.
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The Stoics were the first to develop an explicit theory of propositional connectives. An example they used of what now we call modus ponens: If it is day, then it is light. It is day. Therefore, it is light.
The basic unit of Stoic logic is not the term, as in Aristotle, but the proposition.
Modus ponens is a form of valid argument that students in a first logic course say Yes to right off. They see its validity, and it’s as if one had already known it was a valid form of reasoning before seeing it in the course. Not so with material implication, which is a more contrived creature of modern logic.
William O, John Stuart Mills is the one most famous for giving an empiricist answer to how we come to know logical principles such as modus ponens. The logical empiricists (also called logical positivists) thought such principles are somehow gotten from experience, though it could not be gotten as Mills proposed. Aristotle and Rand/Peikoff also suppose learning of basic logical principles must come from experience (in the present life of an individual), but not in the empiricist way of Mills or in the experiential ways we learn unsupported bodies fall or learn that we have receptors in the skin that guage coolness or warmth by rate and direction of heat transfer.
I do not think that questions of origins of any of the various sorts of knowledge that individuals attain, from infancy to a first course in geometry, can be answered in the finest way without assimilation of the pertinent results of cognitive developmental psychology of the last 70 years.

“If p, then q” is taken in logic texts to be identically equivalent to “Not (p and not-q).” “Not (there is a naturally evolved bird with talons, and it is not a bird of prey)” is identically equivalent to “If there is a naturally evolved bird with talons, then it is a bird of prey.” It seems that we know up front that this “identically equivalent” relation holds however much our knowledge of birds increases; it cannot be found false. Whether there are presently unknown conditions under which this particular “If p, then q” can be found false is open, though until specific prima facie plausible conditions of that sort are proposed (at least in a sketchy way), that open possibility is a vacuous possibility, a degenerate, impotent sort of possibility, whether the if-then concerns nature or mathematics. The nature of birds is a matter of identity, but it seems a wider sort of identity than that in the “identically equivalent” relation. And the latter would seem to be something one learns about later than the former, although maybe the latter is already present in a precursor way in prelinguistic action schemata (eg. there’s more than one way to get attention, more than one form under the schema get attention).
In his book How We Know, Harry Binswanger takes syllogistic inference to be a case in which what is already implicit in the premises is drawn out and made explicit in the conclusion. That is a common perspective on deductive inference. The syllogism is a form of “If p, then q” in which p is a conjunction of two propositions: “If r and s, then q.” For r and s to be true and to bear implicit truths, of course, r and s both have to express awareness of facts (254–55). This viewpoint is smooth with the views of Rand that logic is a form of identification and that existence is identity.
In his book Objectivism: The Philosophy of Ayn Rand, Leonard Peikoff remarked: “The method of logic . . . does reflect the nature and needs of consciousness. It also reflects the other factor essential to a proper method: the facts of external reality. The principle which logic provides to guide man’s mental steps is the fundamental law of reality” (120–21). There are no contradictory facts in reality, I should add, to be thought in conjunction if thought is aimed at fact. To put forth without evidence or design for evidence the thesis that there are naturally evolved birds with talons that are not birds of prey contradicts evident facts without resolving the purported contradiction with other (not-adduced) evident facts. I suggest that denials of modus ponens should be understood as that sort of denial under the basic conception of logic in Objectivism. Logical validities are never independent of all facts of reality.
Some excerpts from Nathaniel Branden’s lectures The Basic Principles of Objectivism: “Logic is the tool of reason. Logic is based on facts, on the fact that that which is, is; but it is not a science of facts. It is a science of method (75).” “One proves a proposition by demonstrating that it is logically necessitated, that its denial would contradict facts already known to exist. . . . . “Until one has grasped that A is A, and that contradictions cannot exist, there can be neither proof nor the concept of ‘proof’. . . . “The Law of Identity is a genetic root of the concept of ‘proof’. . . . (73, transcription in The Vision of Ayn Rand)

2046,
I’ll hold off remarking on pragmatism until we get to Dewey and Lewis.
Concerning the classical ontologists, “they regarded the laws of logic as themselves matters of fact (i.e. ontological in character, not ‘mere’ matters of fact)” (Peikoff 1964, 13).
The classical philosophers basing logic in ontology (such as Plato, Aristotle, Aquinas, and Leibniz) would want to have PNC both as an ontological fact of the world and as a norm, a consciously followed constraint, for ascertaining any fact, whether itself or other facts, whether facts empirical or mathematical. With the variations in ontology between various theories basing logic in ontology are variations in what is ontological form. I think it is always what philosophers say about the ontology of form that is key to their ontology of PNC and their account of how PNC is also a norm.
Below is Peikoff’s representation of Aristotle’s ontology at work in a syllogistic inference. I should like to mention that this text is my personal favorite in Peikoff’s dissertation. Also, I’d like to mention that, as Jonathan Lear showed from Prior Analytics, the certitude of the validity of the syllogism below, and the other first-figure ones, is the base certitude of validity by which Aristotle, using some self-evident logical conversions, certified validity of the syllogisms of the other figures. Lastly, in their lectures and writings concerted with Rand; Branden and Peikoff point to contradictions that occur if one denies the conclusion of this syllogism below while affirming its premises. It is a good assignment for the future to work out the moments of Aristotelian form in rendering those contradictions.
Under Aristotle’s account, we learn the truth of PNC by observing instances of it and performing an intuitive induction to it (also called an abstractive induction). PNC has to be a law prior to the operation of thought in order to be discovered by such observation and abstraction. The normativity of PNC in Aristotle’s account is from the purpose of thought, which is the comprehension of existence. To serve as guide to that purpose in the way PNC serves, PNC must, in Aristotle’s view, be a first principle in existence. We must not think a thing has and has not a certain character at the same time because, as Joseph puts it, “we see that a thing cannot have and not have at once the same character; and the so-called necessity of thought is really the apprehension of a necessity in the being of things’” (Peikoff 1964, 162).
I’ll be looking at Dewey’s expansive notion of logic in turn when we come to it in this series. Looking also at Lewis and at Peikoff’s extractions from both of them. I don’t expect to take up Wittgenstein, and Peikoff also did not. But I thought I’d mention just now a book from Penelope Maddy The Logical Must – Wittgenstein on Logic (2014).